Suppose we want to write an expression stating that $x$ satisfies the property $f(x) \geq f(x'), \forall x' \in X$. A fairly common way of doing this in optimization and economics literature is $$ x \in \arg\max_x \{f(x) : x \in X\}$$ Note that the $x$ under $\max$ indicates the "decision variable," i.e., that you can change $x$ but not any of the other parameters of $f(\cdot)$, while the text $f(x)$ before the colon indicates the "criterion," i.e. the function to be maximized according to the usual ordering $\geq$ of the real numbers.
(If the maximum is known to be unique, we can replace $\in$ with $=$, and there is room for disagreement regarding whether the $x$ under $\max$ should be centered under the whole of $\arg \max$ instead, but the present question does not concern these issues.)
Now suppose that instead of using $f(x)$ and the ordering of the real numbers, we have some other ordering, call it $\succsim$, of the vectors in $X$. What is the best way to denote that $x$ has the property $x \succsim x', \forall x' \in X$?
Answers backed by style guides or examples from published papers are especially welcome.
In my view, the following notation is most consistent with the one above: $$ x \in \arg\max_x \{\succsim : x \in X\}$$ But I often see something like the following notation used instead: $$ x \in \arg\max_{\succsim} \{x \in X\}$$ I don't like this notation, because it puts the "criterion" where I expect to see the "decision variable," but there may be an argument for mentioning the alternative ordering $\succsim$ before the decision variable $x$. Also, it is unclear how to adapt this notation to the case where $X$ is the whole of the real numbers. $\arg \max_{\succsim}$ looks unfinished.